Verify Rolle’s Theorem for the function:f(x) = x2 on [- 1, 1]

1.	Verify Rolle’s Theorem for the function:f(x) = x2 on [- 1, 1]
Answer» We know that (i) f(x) = x² is a polynomial which is continuous for all x ϵ R Hence, f(x) = x² is continuous on [- 1, 1] (ii) f’(x) = 2x exist in [- 1, 1] Hence, f(x) = x² is differentiable on (- 1, 1) (iii) We know that f(- 1) = (- 1)² = 1 Similarly f(1) = 1¹ = 1 Here f(-1) = f(1) The conditions of Rolle’s Theorem are satisfied. There exist at least one c ϵ (-1, 1) where f’(c) = 0 2c = 0 which gives c = 0 We know that value of c = 0 ϵ (-1, 1) Therefore, Rolle’s Theorem is satisfied.

Answer»

We know that

(i) f(x) = x² is a polynomial which is continuous for all x ϵ R

Hence, f(x) = x² is continuous on [- 1, 1]

(ii) f’(x) = 2x exist in [- 1, 1]

Hence, f(x) = x² is differentiable on (- 1, 1)

(iii) We know that

f(- 1) = (- 1)² = 1

Similarly f(1) = 1¹ = 1

Here f(-1) = f(1)

The conditions of Rolle’s Theorem are satisfied.

There exist at least one c ϵ (-1, 1) where f’(c) = 0

2c = 0 which gives c = 0

We know that value of c = 0 ϵ (-1, 1)

Therefore, Rolle’s Theorem is satisfied.

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